High-fidelity quantum information transmission using a room-temperature nonrefrigerated lossy microwave waveguide

Quantum microwave transmission is key to realizing modular superconducting quantum computers and distributed quantum networks. A large number of incoherent photons are thermally generated within the microwave frequency spectrum. The closeness of the transmitted quantum state to the source-generated quantum state at the input of the transmission link (measured by the transmission fidelity) degrades due to the presence of the incoherent photons. Hence, high-fidelity quantum microwave transmission has long been considered to be infeasible without refrigeration. In this study, we propose a novel method for high-fidelity quantum microwave transmission using a room-temperature lossy waveguide. The proposed scheme consists of connecting two cryogenic nodes (i.e., a transmitter and a receiver) by the room-temperature lossy microwave waveguide. First, cryogenic preamplification is implemented prior to transmission. Second, at the receiver side, a cryogenic loop antenna is placed inside the output port of the waveguide and coupled to an LC harmonic oscillator located outside the waveguide. The loop antenna converts quantum microwave fields to a quantum voltage across the coupled LC harmonic oscillator. Noise photons are induced across the LC oscillator including the source generated noise, the preamplification noise, the thermal occupation of the waveguide, and the fluctuation-dissipation noise. The loop antenna detector at the receiver is designed to extensively suppress the induced photons across the LC oscillator. The signal transmittance is maintained intact by providing significant preamplification gain. Our calculations show that high-fidelity quantum transmission (i.e., more than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$95\%$$\end{document}95%) is realized based on the proposed scheme for transmission distances reaching 100 m.

www.nature.com/scientificreports/ transmitted quantum state, with the advantage of extending the interaction over distances. For instance, similar to the case of the cryogenically refrigerated microwave waveguide in Ref. 20 , the transmitted quantum state can establish a remote entanglement of distant qubits. Hence, the proposed scheme enables the use of a microwave waveguide operating at room temperature to connect two superconducting quantum circuits housed in distantly separated cryostats. This approach has the potential to realize a modular superconducting quantum computer (or a local quantum network) without transduction or waveguide cooling. While the purpose of the current work is to provide a viable scheme for coherent transmission, the functionality of the proposed system is another aspect that is beyond the scope of this work. Here, we would like to briefly present the physical mechanism of our proposed system in three steps. First, the quantum microwave fields are pre-amplified at cryogenic temperature before transmission. Second, the microwave fields are transmitted using a room-temperature waveguide. Third and last, the microwave fields at the receiving port of the waveguide (which contain both the signal photons and the noise photons) are intensively suppressed (using an engineered cryogenic loop antenna detector). It then follows that the number of noise photons at the receiver can be controlled and made to less than one. The number of the signal photons at the receiver can be maintained approximately intact owing to the cryogenic preamplification. The organization of the paper is as in the following. In "System", the model of the proposed transmission system considering a lossy room-temperature waveguide is developed. The associated fields and the equation of motion are derived in subsection (A). Subsection (B) presents the quantum signal and noise operators. The system's performance is evaluated in "Performance evaluations". The effective noise temperature and the qubit transmission-fidelity are modeled and calculated considering a typical aluminum waveguide. Finally, "Discussion" discusses the future potential of the proposed system and highlights its economic and practical advantages.

System
Consider a system of two separated superconducting quantum nodes (transmitter and a receiver) placed in two distant dilution refrigerators. The two nodes are connected by a nonrefrigerated lossy microwave waveguide, as shown in Fig. 1. Here, W and h are the width and the height of the waveguide along the x and y axes, respectively.
A. Propagating fields. A typical rectangular microwave waveguide is implemented for signal transmission. The waveguide dimensions can be designed to support single-mode operation. In a rectangular microwave waveguide, the transverse electric field TE 10 mode has the lowest cut-off frequency. Hence, in this work, by having proper waveguide dimensions (see "Methods D"), we consider a TE 10 single-mode operation. The associated electric and magnetic fields can be expressed as follows 30 : where A is the complex amplitude of the TE 10 mode, Z F = 377 µ r ǫ r represents the impedance of the filling material, and � = ω ω c . Here, ω is the microwave signal frequency; ω c = 2πc 2W √ ǫ r denotes the cut-off frequency; µ r and ǫ r are the relative permeability and permittivity of the filling material, respectively; β represents a propagation constant; and c is the speed of light in vacuum.
The classical Hamiltonian of the TE 10 mode is given by denotes the waveguide volume and l is the waveguide length. The propagating microwave field can be quantized through the following relation: where â represents the annihilation operator of the TE 10 mode, ϕ = 2ǫ 0 ǫ eff , and ǫ eff = ǫ r − π 2 c 2 W 2 ω 2 denotes the effective permittivity of the waveguide. Thus, the quantum Hamiltonian is given by Ĥ = ωâ †â .
The equation of motion can be obtained by substituting the quantum Hamiltonian into the Heisenberg equation ∂â ∂t = i [Ĥ,â] . Incorporating the waveguide dissipation and the dissipation-fluctuation noise into this equation, yields ∂â ∂t = −iωâ − Ŵ 2â + √ Ŵf L . Consequently, by using the rotation approximation and setting â =ûe −iωt , the equation of motion is given in the following form: where Ŵ = αv g is the decay time coefficient, α = 2R s 377h µr ǫr h W ( ωc ω ) 2 +0.5 √ 1−( ωc ω ) 2 denotes the attenuation coefficient, and v g = c 1 − ( ω c ω ) 2 represents the group velocity. Here, R s = ωµ m σ , σ and µ m are the surface impedance, the conductivity, and the permeability of the waveguide's metal material, respectively. The quantum noise operator www.nature.com/scientificreports/ f L (t) obeys to the following correlation relation: �f L (t 1 ) † f L (t 2 )� = N th δ(t 1 − t 2 ) , where N th = 1 exp( ω/k B T)−1 , is the plank's constant, k B represents the Boltzman constant, and T denotes the waveguide temperature.

B. Quantum signal and noise.
A quantum source hosted in a dilution refrigerator at the transmitter generates a signal with an annihilation operator ŝ . The signal is preamplified at the cryogenic temperature before being transmitted, as shown in Fig. 1. Amplifying quantum microwave fields have been reported, tested, and characterized for a quite number of years now 31,32 . For instance, different mechanisms for quantum microwave field amplification have been reported. These include using Josephson traveling wave parametric amplification [33][34][35] , metamorphic high-electron-mobility transistors 36 , inelastic Cooper-pair tunneling 37 , and nanomechanical resonators 38 , just to mention possible examples. Importantly, a quantum amplifier can be phenomenologically characterized by its gain and noise factor (see "Methods A"). In this work, we utilize a cryogenic amplifier with gian G and noise factor F a . The amplifier is coupled to the microwave waveguide with a coupling coefficient K. Hence, the annihilation operator of the TE 10 mode at the input port of the waveguide can be calculated using the input-output relation 16,39 , given by (see "Methods B"): where G and F a are the gain and the noise factor of the cryogenic preamplifier, respectively, and f s is the sourcegenerated noise operator. The source noise operator is characterized by zero average f s = 0 and associated noise photons described by N s = f † sf s = 1 2 coth( ω 4πk B T s ) 31,32 , where T s is the cryogenic source temperature. By substituting the input field operator in Eq. (5) into the governing equation of motion in Eq. (4), the field operator û(t) at the output port of the waveguide can be expressed as: where τ = l v g is the interaction/propagation time, and l is the waveguide length. Note that the noise operators f s and f L are uncorrelated.
The solution in Eq. (6) can be used in conjuction with the noise properties to determine the number of photons of the TE 10 mode at the output port of the waveguide (see "Methods C"): The first term in Eq. (7) is the number of TE 10 signal photons, the second term includes the source-generated noise photons and the added preamplification noise photons, and the last term is the dissipation-fluctuation induced noise photons. As can be observed from Eq. (7), the noise contribution at the receiver is significant. Hence, coherent signal transmission requires implementing a scheme for noise suppression. This task is challenging without waveguide cooling to the cryogenic temperatures.
To overcome this dilemma and suppress the noise photons yet without any waveguide cooling, we propose the following approach. A superconducting loop antenna is used inside the waveguide output port and subjected to the TE 10 flux. An LC harmonic oscillator placed outside the waveguide is coupled to the loop antenna, as schematically demonstrated in Fig. 1. The loop antenna induces a voltage across the coupled LC harmonic oscillator based on Faraday's law of induction. The inductance of the loop antenna, which is given by , is included within the inductance of the LC circuit. Here, , and the dimensions d, W r and h r are the thickness, width, and the height of the loop antenna, respectively. The antenna's geometry and the LC components are designed to suppress the induced photons. It then follows that the noise photons can be significantly suppressed to levels obtained in a cryogenically refrigerated waveguide. However, the number of the signal photons can be maintained intact by providing cryogenic preamplification at the transmitter. It is essential to note that the added preamplification noise photons experience the waveguide attenuation and loop antenna suppression and thus have a limited contribution. Thermal occupation of the waveguide and the dissipationfluctuation induced noise photons are also suppressed at the received. On the contrary, while the source-generated noise photons are attenuated and suppressed, they are equally experiencing a preamplification gain. Nevertheless, the source-generated noise photons are reduced by cooling the source at the transmitter to cryogenic temperatures. Classically, the induces voltage across the LC circuit is governed by the Faraday's law of induction, given by: A is the flux to which the loop antenna is subjected and � ∂A = ∂x ∂y� e z represents the differential element of the enclosed area of the antenna. Here, W r and h r are the width and the height of the superconducting loop antenna along the x and y axes, respectively.
By using the expression of the field H in Eq. (2), the induced voltage across the LC circuit can be described by: where V I = iµ 0 µ r Ah r W r . The voltage in Eq. (9) can be quantized using the following relation: where b is the annihilation operator of the voltage in the LC harmonic oscillator; C is the capacitance, satisfying ω = 1

√ LC
; and L is the inductance. The quantization relations given in Eqs. (3) and (10) can be used to obtained a direct relation between the annihilation operators of the TE 10 mode and the LC voltage: Using the expressions in Eqs. (6) and (11), and by writing the input-output relation at the receiver port of the waveguide (see Methods B), the induced voltage b across the LC circuit can be given by: is the loop antenna coupling coefficient, and f w is the thermal occupation of the waveguide, which obeys the characterizations . It then follows that the number of induced photons across the LC harmonic oscillator can be given by (adopting same steps in Methods C): The first term in Eq. 13 corresponds to the number of induced signal photons (denoted by M s ), and the sum of the second and third terms corresponds to the number of induced noise photons (denoted by M n ). Note that the antenna dimensions can be designed to obtain a proper value for κ that sufficiently suppresses the induced noise photons. The signal photons can be maintained significant by providing proper gain. We note here that the main advantage of utilizing the loop antenna at the receiver, instead of a typical coax transition, is leveraging the property of geometrical control of the coupling parameter.

Performance evaluations
A practical room temperature aluminum waveguide of conductivity σ = 3.5 × 10 7 S/m is considered in the numerical estimations presented in this work. The waveguide dimensions are designed to support single mode propagation of the fundamental TE 10 mode. For example, for a waveguide, W = 2.8 cm width and h = 1.4 cm height, only the TE 10 is the propagating mode for the frequency band from 6 to 11 GHz. The waveguide width and the W h ratio can be adjusted to attain desired operation bandwidth (see "Methods D"). Cryogenic amplification for the superconducting quantum circuits placed in dilution refrigerators is typically composed of two stages 6 . The first stage is conducted at a cryogenic temperature of a few mK using a travelling wave parametric(TWPA) amplifier. The second stage is conducted at cryogenic temperature of a few K (e.g., 3 K) using a high-electron-mobility transistor (HEMT) amplifier. Considering cascaded TWPA and HEMT amplifiers with gains of G T and G H , respectively, and noise factors of F T and F H , respectively, the preamplification gain is G = G T .G H , and the noise factor is F a = F T + F H −1 G T . Normally, the noise factors are expressed in terms of the noise temperatures through the relation F ζ = 1 + Here, ζ ∈ {T, H} denote the TWPA and HEMT amplifiers, T N ζ denotes the noise temperature, and T I ζ denotes the input noise temperature. The input noise temperature of the two amplifiers is µ 0 µ r ωh r W r . www.nature.com/scientificreports/ The signal transmittance and the output signal-to-noise ratio (across the LC harmonic oscillator) can be obtained from Eq. (13), as follows: Figure 2a shows the signal transmittance versus the preamplification gain for different waveguide lengths and coupling coefficient κ . Here, ω = 10 GHz. As can be seen, the system transmittance can be maintained unity by having proper preamplification. The number of noise photons induced across the LC harmonic oscillator ( M n ) is shown in Fig. 2b. Here, l = 100 m. The results show that a higher preamplification gain produces smaller induced noise photons. We note that this is provided by having a smaller κ coefficient that maintains an intact transmittance. The output signal-to-noise ratio ( SNR O ) is shown in Fig. 2c for the same parameters and assuming ŝ †ŝ = 40 signal photons. Here, the signal-to-noise ratio is increasing with the preamplification gain, implying a mild impact of the amplification noise on the system performance. In Fig. 2d, SNR O and M n are plotted versus the waveguide length. Here, the same parameters are assumed for a gain of G = 40 dB . The results show that SNR O is independent of the loop antenna parameter κ . However, a smaller κ provides a smaller number of noise (and signal) photons. This can be explained by noting that while the thermal noise photons are intensively suppressed, by having a gain of 40 dB and κ = 5.6 × 10 −4 , the dominating contribution comes from the source-generated noise photons that are only sensitive to the transmittance.
For completeness, we mention that the coupling parameter κ = 5.6 × 10 −4 can be achieved with the following specifications: Capacitance of the LC oscillator of C = 0.5 pF , inductance of the LC oscillator of L = 20 nH , and loop antenna dimensions of W r = 0.27 mm and h r = 0.139 mm. The inductance of the loop antenna with these dimensions is L a = 0.46 nH . The inductance of the loop antenna is typically small and thus included within the inductance of LC harmonic oscillator.
The performance of the transmission system can be determined by calculating the system noise temperature T N Sys , as follows: www.nature.com/scientificreports/ where T I Sys represents the input noise temperature of the system (which is equal to the TWPA input noise temperature T I T ), and F s = SNR I SNR O = F a − N th GKN s + 2N th GKN s e Ŵτ is the system noise factor, where SNR I = �ŝ †ŝ � N s denotes the input signal-to-noise ratio at the source. Figure 3a shows the system noise temperature (normalized against the quantum-limit noise temperature) versus the preamplification gain. Here, the waveguide length is l = 100 m, a unity transmittance is assumed, and T N Q = ω 4π k b is the quantum-limit noise temperature. Interestingly, providing proper preamplification and loop antenna design at the receiver results in near-quantum-limited transmission for the proposed system. For example, while the system noise temperature at 10 GHz is about 10 5 (10 3 ) times the quantum-limit noise temperature for zero (20)dB preamapalification gain, the value is only 15 times the quantum-limit noise temperature for 40 dB preamplifcation gain. This very interesting finding shows the viability of the proposed system as a quantum interconnector. The simulations in Fig.3b show the normalized system noise temperature versus the waveguide length while considering the same parameters as in Fig. 3a and a premplification gain of 40 dB. One can see that the system noise temperature is higher for lower frequencies. This is an expected observation as the thermal noise occupation is inversely proportional with frequency. However, counterintuitively, the system noise temperature at 15 GHz exceeds the system's noise temperature at 10 GHz for waveguide lengths more than 80 m. We note here that the attenuation factor for 15 GHz is greater than that of the 10 GHz. Hence, for a large enough waveguide length, the propagation losses are significant, causing the noise temperature of the 15 GHz to surpass that of the 10 GHz noise temperate.
To further evaluate the system performance, we consider a single quantum bit of information that is generated by the cryogenic source in Fig. 1 and launched toward the room-temperature waveguide for transmission through the proposed system. The generated quantum state at the source is given by 40,41 : where |0� and |1 ξ � =ŝ † |0� are the vacuum and wave packet states, respectively, |a| 2 + |b| 2 = 1 , and ξ(ω) is the spectral profile of the generated wave packet. The source-generated thermal noise f s is uncorrelated with the signal ŝ . Hence, the corresponding source density matrix is given by: The output density operator of the corresponding quantum state across the LC harmonic oscillator is obtained by following the same steps implemented in the previous session to derive Eqs. (12) and (13), yielding: www.nature.com/scientificreports/ where F ι (ω) is the frequency spectrum of the f ι (t) operator, obeying �F † ι (ω ′ )F ι (ω)� = 2πN ι δ(ω ′ + ω) , and ι ∈ {s, L, w} . The first two terms in Eq. (19) correspond to the qubit information generated by the source, the third term corresponds to the source-generated noise and the preamplification noise, the fourth term corresponds to the fluctuation-dissipation generated noise, and the last term corresponds to the thermal occupation noise of the waveguide. The closeness of the input and output quantum states in our proposed transmission systems can be measured by calculating the fidelity between the quantum state at the source and the quantum state across the LC harmonic oscillator at the receiver (which is hereafter refereed to as the transmission fidelity D t ) as follows 42 : where tr() is the trace operator. Figure 4 presents the transmission fidelity (averaged over all possible Bloch sphere states). Here, a transmittance of unity and three different microwave frequencies are considered.
In Fig. 4a the fidelity is presented versus the preamplification gain. Interestingly, suitable preamplification can produce fidelity close to unity (greater than 95% ) for waveguide lengths up to one hundred meters. As expected, it can be observed that the fidelity is increasing for higher frequencies. However, for gain ranges greater than 25 dB, the fidelity for the 5 and 10 GHz frequencies are showing greater values than that of the 15 GHz. We note that for low gain ranges, the suppression at the receiver is partial ( i.e., a small gain designates a large κ ), leading to thermally limited operation. In contrast, large gain ranges lead to substantial suppression (owing to the associated small κ values) causing a suppression limited operation. Nevertheless, higher frequencies are suffering from greater attenuation. Thus, for a unity transmittance, a higher frequency requires relatively smaller κ . Figure 4b, presents the fidelity versus the waveguide length. A preamplification gain of 40 dB is considered. One can observe that fidelity is thermally-limited (i.e., lower microwave frequency experience lower transmission fidelity) for low waveguide lengths less than 29 m, and suppression-limited (i.e., higher microwave frequency experience lower transmission fidelity) for waveguide lengths greater than 60 m. Here, same notes can be mentioned regarding the thermal occupation and the suppression dominance for different frequencies. The simulations in Fig. 4 suggest www.nature.com/scientificreports/ that the preamplification noise is of a mild impact (i.e., increasing gain results in a higher fidelity). We explain this by noting that the preamplification noise photons are prone to waveguide attenuation and receiver suppression. Finally, it is worthy to point out that the source-generated noise photons are proportional to the transmittance, and thus their impact is mitigated only by cooling the transmitter.

Discussion
In quantum microwave transmission systems, the transmission fidelity is severely degraded at room temperature due to significant thermal occupation within the microwave frequency spectrum. Thus, a waveguide cooled to cryogenic temperatures has been proposed to mitigate thermal occupation and achieve acceptable transmission quality. Other techniques, such as implementing controllable coupling between qubits (at the two sides of the link) and the connecting channel, have also been proposed. The potentials and limitations of these techniques were briefly discussed in the introduction.
In this study, we propose a novel technique to achieve reduced noise levels without waveguides refrigeration simply by designing a suitable loop antenna at the receiver side. For example, in Fig. 4, the thermal waveguide occupation at room temperature is N th = 1.22 × 10 3 for a microwave frequency of ω 2π = 5 GHz. However, the number of the induced noise photons across the LC harmonic oscillator is only κN th = 0.413 , which is equivalent to that for the same waveguide cooled to 0.2 K. Same calculations for ω = 10 GHz, with N th = 609 and κN th = 0.34 , corresponds to noise level equivalent to a cooled waveguide at 0.35 K. Combining this proposed loop antenna technique with proper cryogenic preamplification results in a room-temperature lossy waveguide with high-fidelity transmission (above 95% ) over significant transmission distances (up to 100 m).
To the best of our knowledge, this study is the first proposal of a high-fidelity microwave transmission system using a typical lossy nonrefrigerated waveguide. Our approach has the potential to realize a modular quantum computer with waveguides placed outside dilution refrigerators. This feature is very important because scaling quantum computers up to a capacity of thousands of qubits is crucial for leveraging quantum supremacy. The state-of-the-art capacity of current superconducting quantum computers is less than 127 qubits 15,21,43,44 , and boosting the number of qubits to thousands is very challenging, especially from a heat management standpoint. This difficulty is due to the fact that adding a qubit requires connecting a considerable number of cables and related components, which imposes an overwhelming heating load. For example, the estimated cost per qubit to maintain the required cryogenic refrigeration and connect the pertinent cables is approximately $10 K 45 . Thus, a scaled quantum computer utilizing a giant dilution refrigerator is expected to cost tens of billions of dollars 46 . Connecting separated quantum nodes (or processors) by coherent signaling is a promising approach for efficient scaled quantum computation. The findings reported here have the potential to expedite the realization of sought-after modular superconducting quantum computers capable of containing thousands (million) of qubits. Finally, we note that the proposed scheme (cryogenic preamplification at the transmitter and a passive cryogenic detection circuit at the receiver) is technically very easy to implement.

Methods
The preamplifier. Consider a preamplifier with a power gain G and a noise factor F a . The signal and the noise powers at the input of the preamplifier are generated by the source, given by ŝ †ŝ and f † sf s , respectively. The signal power at the output port of the amplifier is G ŝ †ŝ . The noise factor F a is by definition the signal-tonoise ratio at the input to the signal-to-noise ratio at the output of the preamplifier. Hence, the noise power at the output of the preamplifier is given by F a G f † sf s .
The input-output relations. The fields operators at the waveguide ports can be described through the input-out relations. The schematic in Fig. 5 shows the fields operators at the two ports of the waveguide with the corresponding coupling parameters K and κ . It can be seen that the thermal occupation of the waveguide loads the transmitter and receiver nodes with √ Kf w and √ κf w operators, respectively. The thermal load at the transmitter can be extracted by implementing a circulator between the amplifier and the waveguide. However, the thermal load at the receiver can be suppressed by controlling the coupling parameter κ . The operators' input-

Number of photons.
Using the field operator û(t) expression in Eq. (6), and noting that the noise operators f s and f L (t) are uncorrelated, the number of photons at the output of the rectangular waveguide is given by: The expression of the number of photons in Eq. (21) can be simplified by incorporating the noise properties, which are �f † s (t ′ )f s (t)� = N s δ(t ′ − t) and �f † L (t ′′ )f L (t)� = N th δ(t ′′ − t) , yielding the form in Eq.7.
Waveguide dimensions. The waveguide dimensions are designed to support a single mode operation. The cut-off frequency of the propagating modes in a rectangular microwave waveguide is given by ω mn c = 2πc √ ǫ r µ r ( m W ) 2 + ( n h ) 2 , where n = 1, 2, 3, ... and m = 1, 2, 3, ... . The lowest cut-off frequency is associated with the TE 10 mode, given by ω 10 c = 2πc 2W √ ǫ r . The second lowest cut-off frequency is for the TE 01 mode, given by ω 01 c = 2πc 2h √ ǫ r . Hence, the single mode frequency range (named hereafter single mode bandwidth) is given by B w = ω 10 c − ω 01 c = 2πc 2 √ ǫ r µ r W−h Wh . Figure 6 shows the attenuation factor along with the single mode bandwidth for TE 10 mode. In Fig. 6a, the simulations are shown considering different waveguide widths for W h = 1 2 . On the other hand, Fig. 6b considers W = 2.8 cm for different W h ratios. It can be seen that smaller waveguide widths result in larger single-mode bandwidths. However, smaller widths endure larger mode attenuation. Furthermore, for a given waveguide width, the smaller the W h ratio, the larger the single-mode bandwidth. Yet, the experienced attenuation is increasing. Hence, a trade-off takes place between the single-mode bandwidth and the waveguide attenuation.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.